

A199711


Triangular array: T(n,k) gives the number of numerical semigroups of genus n and multiplicity k, (n>=1, k>=2).


1



1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 4, 4, 1, 1, 3, 6, 7, 5, 1, 1, 3, 7, 10, 11, 6, 1, 1, 3, 9, 13, 17, 16, 7, 1, 1, 4, 11, 16, 27, 28, 22, 8, 1, 1, 4, 13, 22, 37, 44, 44, 29, 9, 1, 1, 4, 15, 24, 49, 64, 72, 66, 37, 10, 1, 1, 5, 18, 32, 66, 85, 116, 116, 95, 46, 11, 1
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,5


COMMENTS

A numerical semigroup is a subset S of N, the nonnegative integers, that is closed under addition, contains the element 0 and such that NS is finite. The cardinality of NS is called the genus of S. The least positive integer belonging to S is called the multiplicity of S. The number of numerical semigroups of genus n is A007323(n).


LINKS

Table of n, a(n) for n=1..78.
V. Blanco, P. A. GarciaSanchez and J. Puerto, Computing the number of numerical semigroups using generating functions, arXiv:0901.1228v3 [math.CO], 2009.
Nathan Kaplan, Counting Numerical Semigroups, arXiv:1707.02551 [math.CO], 2017.


EXAMPLE

Triangle begins
.n\k...2....3....4....5....6....7....8....9...10
= = = = = = = = = = = = = = = = = = = = = = = = =
..1....1
..2....1....1
..3....1....2....1
..4....1....2....3....1
..5....1....2....4....4....1
..6....1....3....6....7....5....1
..7....1....3....7...10...11....6....1
..8....1....3....9...13...17...16....7....1
..9....1....4...11...16...27...28...22....8....1
...
T(3,3) = 2: The two numerical semigroups of genus 3 and multiplicity 3 are S = N  {1,2,4} and S = N  {1,2,5}.


CROSSREFS

Cf. A007323 (row sums).
Sequence in context: A181322 A004070 A180562 * A048887 A047913 A152977
Adjacent sequences: A199708 A199709 A199710 * A199712 A199713 A199714


KEYWORD

nonn,tabl


AUTHOR

Peter Bala, Nov 09 2011


STATUS

approved



